Let $q(x_1,\dots, x_6)$ be a arbitrary multivariate density that we choose beforehand. What I want is to calculate a new multivariate density $p(x_1, \dots, x_6)$, which is obtained by minimizing the ...

In the Feynman lectures on physics, Feynman in talking about the principle of least action, discusses how we should be able to find the true path $x(t)$ which has the least action, and the way to do ...

In the Calculus of Variations book by Gelfand and Fomin it says to consider the transformation
$$x^{*} = \Phi(x,y,y')$$
$$y^{*} = \Psi(x,y,y').$$ Here it seems that $y'$ is the derivative of $y$ with ...

My functional is $J[f] = \int_{-\infty}^{\infty} f(x) \log f(x)\,dx$. I want to maximize it using the calculus of variations.
In order to use the Euler-Lagrange equation, I define $L(t, y, y')$ such ...

Given the arc-length of a parametric curve, $\int_a^b\|\gamma'(t)\|$ if the parametric curve was non-convex, can the arc length be a convex function?If the parametric curve was convex, will the arc ...

The fundamental lemma of the calculus of variations is often presented as:
If $M(x) \in C[a,b]$ such that $\int_{a}^{b}{M(x)\eta(x)} = 0 ~~\forall\eta\in C^1[a,b],\eta(a)=\eta(b)=0$, then $M(x)=0$ for ...

An infinitisemally small ball is placed at the top of a ramp which has a height of 1m and ends 1m away horizontally. What is the optimal curve of the ramp to minimize time taken for the ball to reach ...

Suppose I have a functional L. For example $L = y+3y'$. Where y is itself a function of real variable x
It's easy for me to evaluate the Functional Derivative of L via the Euler Lagrange Equations:
...

We have a set of locally finite perimeter and a sequence of sets $\{E_h\}_h$ with $C^1$ boundary such that $$E_h\to E \text{ and } \mu_{E_h}\stackrel{*}{\rightharpoonup} \mu_E,$$ where $\mu_{E_h}$ and ...

Suppose I wish to minimise the integral
$$I = \int_{s_0}^{s_1}\int_{t_0}^{t_1}F\, dt ds$$
Where $F$ is a function of the six variables $x(s,t)$, $y(s,t)$, and their four partial derivatives, ie
$$F ...

Let $D\subset\mathbb R^n$ be open and bounded. Consider $\Delta u=f$ in $\Omega$ and $u=g$ on $\partial\Omega$. Let $g\in W^{1,2}(D)$ and $f\in L^\infty(D)$.
Then the minimizer of
$$
I(u)=\int_\Omega ...

I'm reading Howell's Applied Solid Mechanics to gain background for a research project. I'm struggling with the following derivation in the text that the authors refer to as a "lengthy exercise." The ...

I completely understand the proof for the Euler-Lagrange equation for a general function $F(x,y,y')$. However, when I try to use the same proof technique on a function $F(y')$, I run into a curiosity ...